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Complete characterization of $2$-near perfect numbers with exactly 2 prime factors

This paper provides a complete characterization of 2-near perfect numbers with exactly two prime factors by proving the non-existence of such odd numbers and classifying all even instances of the form $2^kp^m$ (with $m \ge 3$ ) into a specific family.

Original authors: Richard Fearon, Henry Foushee, Benjamin Porosoff, Alexander Skula, Joshua Zelinsky, Kyle Zhang

Published 2026-05-26

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CC BY 4.0

Original authors: Richard Fearon, Henry Foushee, Benjamin Porosoff, Alexander Skula, Joshua Zelinsky, Kyle Zhang

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a giant puzzle made of numbers. In the world of mathematics, there's a special kind of puzzle piece called a Perfect Number. A perfect number is a number where if you add up all its "building blocks" (its divisors), the total equals exactly twice the number itself. For example, the number 6 is perfect because its divisors are 1, 2, and 3, and $1+2+3 = 6$ (which is half of 12, or $2 \times 6$ ).

But what if the sum of the building blocks is almost twice the number, but just a little bit off? That's where this paper comes in.

The "Near Perfect" Concept

The authors are studying 2-near perfect numbers. Think of these as numbers that are "almost" perfect, but they are missing exactly two specific building blocks from the final sum.

If you take a number $n$ , add up all its divisors, and the total is equal to $2n$ plus two extra divisors ( $d_1$ and $d_2$ ), then $n$ is a 2-near perfect number.

The Analogy: Imagine you are trying to fill a bucket that holds exactly $2n$ liters of water. You pour in all the water from the divisors, but the bucket overflows by exactly two specific cups of water ( $d_1$ and $d_2$ ). The paper asks: "Which numbers cause this specific kind of overflow?"

The Specific Mystery: Two Prime Ingredients

The paper focuses on a very specific type of number: those made of exactly two distinct prime ingredients.

One ingredient is the number 2 (the "even" part).
The other ingredient is an odd prime number (like 3, 5, 7, etc.), raised to some power.

The authors wanted to know: If we have a number made of 2 and an odd prime, and it's 2-near perfect, what does it look like?

The Big Discovery

Previous researchers had solved the puzzle for cases where the odd prime appeared only once or twice (like $2^k \times p$ or $2^k \times p^2$ ). They found a few specific numbers.

However, there was a lingering guess (a conjecture) that if the odd prime appeared three or more times (like $2^k \times p^3$ ), there would be no such numbers at all, or only a tiny, finite handful.

This paper proves that guess wrong.

Here is the simple breakdown of their findings:

The "Infinite" Family: They discovered that there is actually a whole family of these numbers that goes on forever (if a certain famous math mystery about "Mersenne primes" is true).
The Recipe: Every single 2-near perfect number with two prime factors (where the odd prime appears 3 or more times) follows this exact recipe:
- It is made of a power of 2 ( $2^k$ ).
- It is multiplied by a specific type of odd prime called a Mersenne prime (a prime that looks like $2^{k+1} - 1$ ).
- The odd prime must be cubed (raised to the power of 3).
- The Formula: The number is $2^k \times (2^{k+1}-1)^3$ .
The Missing Pieces: For these numbers to work, the two "extra" divisors that cause the overflow are always the prime itself ( $p$ ) and the prime squared ( $p^2$ ).

Why This Matters (In Math Terms)

The authors didn't just find one or two examples; they proved that this is the only way these numbers can exist.

If you try to build a 2-near perfect number with two prime factors where the odd prime appears 3 or more times, it must fit this specific pattern.
If there are infinitely many Mersenne primes (a widely believed but unproven fact in math), then there are infinitely many of these 2-near perfect numbers.

The "Odd" Case

The paper also briefly addresses numbers made of two odd primes (no 2 involved). They proved a simple fact: You can't make a 2-near perfect number with two odd primes. It's mathematically impossible, like trying to build a square circle.

Summary

Think of the authors as detectives solving a case of "Almost Perfect Numbers."

The Suspects: Numbers made of 2 and an odd prime.
The Crime: Being "2-near perfect" (sum of divisors = $2n$ + 2 extra pieces).
The Verdict: If the odd prime appears 3 or more times, the suspect must be of the form $2^k \times p^3$ , where $p$ is a special Mersenne prime. The "missing pieces" are always $p$ and $p^2$ .

They also checked the "deficient" version of the problem (where the sum is less than $2n$ by two pieces) and found a different, simpler set of rules for those.

In short, they closed the book on this specific type of number, showing that while the list might be infinite, the pattern is rigid and predictable.

Technical Summary: Complete Characterization of 2-Near Perfect Numbers with Exactly 2 Prime Factors

Problem Statement
The paper addresses the classification of 2-near perfect numbers, defined as positive integers $n$ satisfying $\sigma(n) = 2n + d_1 + d_2$ , where $d_1$ and $d_2$ are distinct positive divisors of $n$ . While the cases for $k=1$ (1-near perfect) and $k \ge 4$ ( $k$ -near perfect) are well-understood, the cases for $k=2$ and $k=3$ remain more open. Specifically, the authors focus on numbers of the form $n = 2^k p^m$ , where $p$ is an odd prime and $m \ge 3$ . This work completes the classification for all 2-near perfect numbers with exactly two distinct prime factors, building upon prior results that handled the cases $m \in \{1, 2\}$ .

Methodology
The authors employ a rigorous number-theoretic approach combining algebraic manipulation of the divisor sum function $\sigma(n)$ , parity arguments, and $p$ -adic valuation analysis. The proof strategy is divided into two primary regimes based on the relationship between $2^{k+1}$ and $p$ :

Foundational Lemmas: The authors establish that no odd 2-near perfect numbers exist with exactly two prime factors, restricting the problem to even numbers $n = 2^k p^m$ . They derive key identities, such as $(2^{k+1} - p)(1 + p + \dots + p^{m-1}) = 1 + d_1 + d_2$ , and analyze the parity of omitted divisors based on the exponent $m$ .
Case I ( $2^{k+1} \ge p^2 + 1$ ): The authors assume this inequality holds and derive a contradiction. They establish lower bounds on the omitted divisors, showing that the larger omitted divisor must be $n/2^j$ (where $2^j < p < 2^{j+1}$ ). Through a series of propositions, they constrain the exponent $m$ , proving it must be odd and eventually forcing $m=3$ . By analyzing 2-adic valuations and divisibility constraints (specifically $p^3 \mid 2^{k+1}-1$ ), they demonstrate that no solutions exist in this regime, even for the specific candidate $n = 2^5 \cdot 3^3$ .
Case II ( $2^{k+1} < p^2 + 1$ ): In this regime, the authors eliminate impossible forms for the omitted divisors. They prove that neither omitted divisor can be a power of 2 and that $p^2$ cannot divide both divisors. By analyzing the exponents of $p$ in the omitted divisors, they deduce that the divisors must be of the form $2^a p$ and $2^c p^d$ . Further algebraic manipulation forces $d = m-1$ and ultimately $m=3$ . Finally, they prove that $p$ must be a Mersenne prime of the form $2^{k+1}-1$ .

Key Contributions and Results
The paper provides a complete classification of 2-near perfect numbers of the form $n = 2^k p^m$ with $m \ge 3$ . The main result is Theorem 3, which states:

If $n = 2^k p^m$ $n = 2^{k} p^{m}$ is 2-near perfect with $p$ $p$ an odd prime and $m \ge 3$ $m \geq 3$ , then:
- $m = 3$ ,
- $p = 2^{k+1} - 1$ (a Mersenne prime),
- The omitted divisors are $p$ and $p^2$ .

This result implies that the existence of infinitely many such numbers is equivalent to the existence of infinitely many Mersenne primes. Consequently, the paper disproves Conjecture 1 (proposed by Aryan et al.), which hypothesized that only finitely many 2-near perfect numbers of the form $2^k p^m$ exist for $m \ge 3$ .

Combined with previous work handling $m=1$ and $m=2$ , this yields a complete characterization of all 2-near perfect numbers with exactly two distinct prime factors.

Significance
The authors state that this work resolves the classification problem for 2-near perfect numbers with two prime factors, a gap left by prior research. By disproving the conjecture of finiteness (conditional on the widely believed infinitude of Mersenne primes), the paper clarifies the density and structure of these numbers. The authors also note that their results have been formally verified using the Lean 4 proof assistant and the Mathlib library, ensuring the correctness of the complex algebraic derivations.

Future Directions
The paper concludes by outlining open problems, including the classification of 2-near perfect numbers with three distinct prime factors, the study of 3-near perfect numbers, and the investigation of "hybrid" forms where divisors are both added and subtracted ( $\sigma(n) = 2n + d_1 - d_2$ ).